Volume asymptotics and Margulis function in nonpositive curvature

Abstract

In this article, we consider a closed rank one $C^\infty$ Riemannian manifold $M$ of nonpositive curvature and its universal cover $X$. Let $b_t(x)$ be the Riemannian volume of the ball of radius $t>0$ around $x\in X$, and $h$ the topological entropy of the geodesic flow. We obtain the following Margulis-type asymptotic estimates \[\lim_{t\to \infty}b_t(x)/\frac{e^{ht}}{h}=c(x)\] for some continuous function $c: X\to \mathbb{R}$. We prove that the Margulis function $c(x)$ is in fact $C^1$. If $M$ is a surface of nonpositive curvature without flat strips, we show that $c(x)$ is constant if and only if $M$ has constant negative curvature. We also obtain a rigidity result related to the flip invariance of the Patterson-Sullivan measure.